Organizers:
Alkis Akritas (akritas@uth.gr)
Natasha Malaschonok
(malaschonok@math-iu.tstu.ru
)
Bill Pletsch
(bpletsch@tvi.cc.nm.us)
Michael Wester (wester@math.unm.edu)
Overview:
Education has become one of the fastest growing application areas for computers in general and computer algebra in particular. Computer algebra tools such as TI-92/89, Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD or Reduce, make powerful teaching tools in mathematics, physics, chemistry, biology, economy.
The goal of this session is to exchange ideas and experiences, to hear about classroom experiments, and to discuss all issues related with the use of computer algebra tools in classroom (such as assessment, change of curricula, new support material, ...)
If you are interested in presenting a paper, please contact one of the organizers.
Papers presented at the session will be considered for publication in a special issue of the Journal of Symbolic Computation titled Applications of Computer Algebra. The call for papers is available. Speakers should also be aware of the ACA'2002 Important Dates.
Talks:
Jerry Uhl
Professor of Mathematics, Professor of Education
University of Illinois at Urbana-Champaign
Today, largely because of technology, math permeates more of university education and life outside the university than ever before. In almost every field of study, math is strikingly more important than it was even twenty years ago. Many fields of study place demands on math that were not even thought about twenty years ago. Today's culture is both math needy and math hungry.
But most students arrive at the university poorly prepared for the math they will use in their university education and their work. They may be prepared for the math their parents and grandparents had to learn, but this is not adequate preparation for the math they will actually use in today's world of technology. In fact, most of the students who score at the top on standardized tests are poorly prepared. Some can even manipulate x's and y's with reckless abandon, but have little understanding of how those skills relate to their university courses and their work beyond the university. They do not realize that because of technology such skills are not as valuable today as they were even twenty years ago. With wise use of technology, the CPM students move from symbol pushing to understanding and see mathematics as worthy of serious attention.
On the other side of this problem is that many of the most creative students have been turned off with math courses in algebra and trigonometry that emphasize drill on rote manipulations of seemingly meaningless symbols. These are often the students who continually ask, "What's this stuff good for?" only to be told (often incorrectly) that they will need it later on. This group needs a new way of learning with content appropriate for answering the question "What's this stuff good for in today's world?"
College Prep Math (CPM) is designed to address both issues. For students who have been enjoying their school math, the course is an opportunity to learn what is important at the university level before they get to the university. For students who have been turned off by school math, this course is an opportunity for a new start and a fast track to the math (in context) that actually arises in science, engineering, technology and in the workplace.
O. V. Lobanova
Glazov, Russia
Much attention is given to work with future mathematics teachers for their preparation for conducting extracurricular work with secondary school students in teacher's training colleges. The author has the experience of preparing students of mathematics faculty for using computer for conducting both ordinary lessons and optional classes on the mathematics and tutorials on mathematics. A special course "Mathematics and computer" was delivered for 4 years where issues of case of spreadsheets and mathematical software packages by teachers of mathematics. Students perform different individual tasks, prepare files which are used when conducting talks and lessons on various themes. For example, tutorials "Remarkable curves", "Methods of optimization", "Methods of defining extremes of functions", "Stages of history of Mathematics", etc. Parts of lessons on various themes were conducted and discussed, e.g. on themes "Approximate methods of solution of equations and systems of equations", "Properties of logarithmic functions", "Trigonometric functions and their properties", etc. The created files are designed for diversified use: some are demonstrated by the teacher when the new material is explained, others are actively handled by pupils, e.g. students paste graphs or just read texts and answer questions. The research has revealed that students like the enumerated activities. They note that during their participation in the seminar their attitude to mathematics school teaching methodology changed. Now they understand more clearly how important it is to choose the proper methodology of explaining the material. When performing individual tasks they have to revive the school mathematics course from a new point of view, work with school textbooks and manuals. Besides, their preparation for conducting extracurricular work on mathematics is enhanced. When choosing the theme students work with journals "Mathematics at school", "Computer science and Education", "Quantum". Many students wish to continue their work on the selected themes. All of them got an opportunity to make reports at students' research conferences. Their reports were delivered with interest. Some students who took part in the seminar conducted term and diploma papers on problems of use of computer by a mathematics teacher. For example, in 2001 students working on their diploma papers conducted series of talks for pupils using computer on different themes, including history of mathematics and entertaining mathematics. These talks have interested pupils. The author's experience is described in the report in details.
Fernando L. Pelayo and Juan A. Aledo
Escuela Politécnica Superior de Albacete
University of Castilla-La Mancha, Campus Universitario
02071 - Albacete. Spain
fpelayo@info-ab.uclm.es and jaledo@pol-ab.uclm.es
and
Juan C. Cortés
Department of Applied Mathematics
Universidad Politécnica de Valencia, Campus Universitario
46071 - Valencia. Spain
jccortes@mat.upv.es
Motivated by the resolution of an open problem on regular polygons, we study how to associate every real x greater or equal than 2, with a regular object which is generated by the infinite iteration of a polygonal (sometimes open) characterized by a certain angle which depends on x.
These regular objects can be classified into three different kinds:
In fact we give an effective algorithm which allows, by means of the Computational Algebraic System Mathematica, to identify in finite time the rational numbers from the real ones.
N. Kouvatsi, E. Maou and P. Vlachos
Computer Science and Mathematics Department
American College of Thessaloniki
Greece
In this paper we present the ways in which the introduction of Computer Algebra Systems (CAS) led to a change in the teaching philosophy and curricula of College level Calculus courses. We review the experiences of various Universities on introducing CAS and we briefly present our own experience of the last six years incorporating CAS into the mathematics curriculum at the American College of Thessaloniki (ACT).
We describe the shift in the teaching approach of College level calculus from a classical mathematical analysis course to a CAS based course, where emphasis is placed on developing critical and analytical thinking skills as well as problem solving skills. Main emphasis is given in the content reform implemented at ACT through specific business applications. We present the change in the role of CAS as a support tool to the main teaching and learning platform. We discuss specific ways to compensate for the overhead introduced to students and instructors alike by teaching the necessary computer skills along with the course material. We finally present our plans for the implementation of a web-enhanced course using webMathematica as the CAS platform.
N. A. Malaschonok
Entire functions form an extensive class of functions of complex variables that takes an active part in numerous applications of complex analysis. If a function f is not a polynomial, then one of its main characteristics is rate of increase of its maximum modulus. Usually there are considered the behaviour of maximum modulus on the circles |z| = r if r approaches infinity (order, loci, etc.) and rate of its increase on the rays arg z = phi (indicatrix). There appears a natural connection with functions associated with respect to Borel.
The subject of our constructions is an algorithmization of necessary estimations for the described activities.
We make the data base D which consists of the values of the pointed characteristics for the main elementary functions. The base D includes as well some estimations for the modulus of these functions on the circles and on the rays.
We define some principles for calculating the characteristics of increase and of necessary estimations and create the catalogue A of standard algorithms.
For a given function using libraries D and A we calculate if necessary:
First of all this program has the applied significance, as the calculating of the characteristics described above is the necessary part of dealing with entire functions.
Above all we intend to use it in teaching -- for training to work with estimations of entire functions.
Burkhard Alpers
FH Aalen, University of Applied Sciences
Germany
balper@fh-aalen.de
The notion of (mathematical) microworld is used to denote a learning environment with computational objects, operations and activities "with the purpose of inducing or discovering their properties, and constructing an understanding of the system as a whole. Experimentation, hypothesis generation and testing, and open-ended exploration are encouraged" (Edwards, 1998, p. 67). In (Alpers, 2002), we investigated the usefulness of Computer Algebra Systems (CAS) for implementing such environments. Since CAS provide already a comprehensive set of mathematical objects, operations and representations, they are obviously "good candidates" for implementation but they need additional programming of higher-level objects and operations to make them more suitable for pupils as (Kent, 2000) already observed. This holds in particular for mathematical microworlds which model an application scenario. The objects and operations offered should then be meaningful within this scenario.
As an example for such an application-oriented mathematical microworld we implemented a learning environment called "Formula 1" within the CAS Maple. The mathematical objects and operations essentially comprise geometric objects for course construction and functions as well as function transformations for modelling motion. In (Alpers, 2002) we showed that from an implementational point of view, it was relatively easy to realize this world taking into account essential properties like multiple linked representations, expressive means of the learner, feedback, adaptability and extensibility. In this talk, we describe and discuss in more detail the learning opportunities provided by this microworld and how they were used in a learning unit with grade 12 pupils.
Alpers, B. (2002): CAS as Environments for Implementing Mathematical Microworlds, Int. Journal of Computer Algebra in Mathematics Education, Fall 2002 (to appear).
Edwards, L. (1998). Embodying Mathematics and Science: Microworlds as Representations, Journal of Mathematical Behavior, vol. 17, no. 1, pp. 53-78.
Kent, P. (2000). Expressiveness and Abstraction with Computer Algebra Software, Journée d'étude: Environnement informatiques de calcul symbolique et apprentissage des mathématique, Rennes, France.
Maher Ahmed
Wilfrid Laurier University
Physics and Computing Department
Waterloo, Ontario, Canada, N2L 3C5
Rabab Ward
University of British Columbia
Electrical and Computer Engineering Department
Vancouver, BC, Canada, V6T 1Z4
Nawwaf Kharma
Concordia University
Electrical and Computer Engineering Department
Montreal, Canada, H3G 1M8
A knowledge-base system for solving mathematical exercises is developed. The system can be used in teaching students a basic mathematical course. The system automatically recognizes the handwritten symbols. Then, understands the questions, interprets the mathematical expressions and finally solving the problems. Two mathematical topics are addressed. The first one is the differentiation and the second one is finding a general term in a series of integers.
The first steps of the recognition stage are scaling, thinning and representing each thinned symbol by a model. Each model consists of several short straight lines. The system recognizes each symbol by comparing its resultant model with the stored ones. After recognizing all symbols, the system applies another set of rules to understand the problem and interpret the expression. Finally, appropriate rules are applied to solve the question whether it is a differentiation problem or finding a common factor in a series of integers.
Igor Gachkov
Karlstad University
Department of Engineering Sciences, Physics and Mathematics
Karlstad, Sweden
Nowadays there are practically no mathematical courses in which Computer Algebra Systems (CAS) programs, such as Mathematica, Maple, and TI-89/92, are not used to some extent. However, generally the usage of these programs is reduced to illustration of computing processes: calculation of integrals, differentiation, solution of various equations etc. This is obtained by usage of standard command of type: Solve[...] in Mathematica. At the same time the main difficulties arise at teaching non-conventional mathematical courses such as Coding theory, Discrete mathematics, Cryptography, Scientific Computing, which are gaining the increasing popularity now. Now it is impossible to imagine a modern engineer not having basic knowledge in Discrete mathematics, Cryptography, Coding theory. Digital processing of signals (digital sound, digital TV) have been introduced in our lives.
The first positive experience of using the opportunities of CAS (in particular Mathematica) was acquired by the author during the development of a course "Coding Theory in Mathematica". During preceding years courses in Coding Theory have been given only for students on the postgraduate level. This is due to the complexity of the mathematical methods used in most of codes, such as results from abstract algebra including linear spaces over Galois Fields. With the introduction of computers and computer algebra the methods can be fairly well illustrated. The author has (in cooperation with Kenneth Hult) developed a course, "Coding Theory in Mathematica", using the wide range of capabilities of Mathematica. The course was given during the autumn 1994 at Jonkoping University, Sweden, on undergraduate level with a minimum of prerequisites. The hands on sessions were based on a package of application programmes/algorithms, developed to illustrate the mathematical constructions, used in coding theory to encode and decode information.
The first results of our courses were presented at the International IMACS Conference on Applications on Computer Algebra in University of New Mexico in 1995, along with a short presentation of the package "Coding Theory" [1]. It is natural, that the best results in mastering courses can be achieved only in the case of good enough technical equipment, in particular availability of programs used in course, presence of computer halls and possibilities of access. However, the increasing interest and popularity of such type of courses complicates the task. Beginning from 10-15 students on course "Coding Theory in Mathematica" during the autumn 1994 at Jonkoping University the number increased to 40-45 students in 1995-97 [2]. Now during the spring 2002 there are 140 participants in the course Discrete mathematics at Karlstad University and the same number students during the autumn 2001.
Simplicity in usage, availability, affordable pricing, possibility of operation in any conditions (in particular in the big audiences without special equipment), compatibility with a PC, big enough computational capabilities have lead to an idea of using TI-89/92/83. Considering all the advantages after comparing Mathematica, Matlab and TI-89/92, attempts to develop a course Discrete mathematics with TI-89/83 was carried out [3]. Funding for idea development 2001 from Chalmers Teknikpark, Teknikbrostiftelsen in Gothenburg, Sweden Project: Boolean algebra with TI-83 (in cooperation with Jorryt van Bommel), has enabled to develop the software package programs and to include it in the course. Now the package is used in the course Discrete mathematics: Logic and Boolean functions and is very popular among the students [4].
The packages ''Boolean algebra'' and ''Set theory''
The packages "Boolean algebra" and "Set theory" are a program packages, which was created especially for TI-83/89 and is used for teaching in course of Discrete Mathematics based on a traditional textbook Ralph P. Grimaldi, Discrete and Combinatorial Mathematics an Applied Introduction, Fourth edition. Actually this package is a natural development and a further edition of the package "Boolean.m" in Mathematica, which has been used by the author during a long time for teaching in this course. The packages "Boolean algebra" and "Set theory" for TI 89 contains programs which display different steps with illustrative explanations calculation in the Boolean algebra and Set theory.
Calculators allow to change the teaching process by replacing of Mathematica with TI-89 due to their safety, low price and because they are easy to use and are possible to develop and provide. Of course Mathematica has more powerful calculating possibilities, but calculators are very flexible, and can therefore be used during lectures in big rooms without technical facilities.
Giovannina Albano, Matteo Desiderio, Giuliano Gargiulo
Dipartimento di Ingegneria dell'Informazione e Matematica Applicata
Centro di Ricerca in Matematica Pura ed Applicata
Universita degli Studi di Salerno, Via Ponte Don Melillo, I-84084
Fisciano, Italy
We devise several Mathematica packages to be integrated into a traditional course on ordinary differential equations. The key idea is supporting conjecture and counterexamples generation through interaction, both numerical and symbolical. Several fundamental topics (existence and uniqueness of solutions, linearisation, stability) are discussed with the help of simulations and interaction, with the explicit aim of stimulating the students to find out "how far" known phenomena actually extend, which variations show up in this extension process and "what goes wrong" when certain hypotheses are dropped. Thus, for examples, the symbolic power of a CAS allows testing several families of ODEs for uniqueness of solutions to IVPs; this helps proving that, while the Lipschitz condition is not strictly necessary for uniqueness, it is a limit point on the scale of Holder continuity. As a result, we expect, and plan to experimentally test, a deeper overall understanding of the subject, with students developing a greater ability to find out links among distinct areas of mathematics.
Bill Pletsch
Albuquerque Technical Vocational Institute
Albuquerque, New Mexico, USA
The use of computer algebra in the mathematics classroom will be discussed. As an example, a specific computer classroom lecture will be demonstrated, that of the behavior of tangent exponential functions. (A tangent exponential is an exponential function that is tangent to a curve.)
Computer algebra is in a unique position to aid students in learning the above concept. A graphing calculator is not enough, since symbol manipulation is required to do the necessary limits. Simultaneously, the presentation will demonstrate by example, the modern methods of presenting a mathematical concept from the numerical, graphical, and symbolic points of view.
Finally, we explore the pitfalls of technology. In particular, how does one deliver technologically sophisticated instruction in a readily accessible form? A solution will be offered: the video capture of all aspects of a presentation: audio and video feed of computer, documents, and instructor to be re-rendered into manifold electronic forms.
Yuzita Yaacob
Zawiyah Mohammad Yusof
Noraini Hassan
Khairina Atika Mohd. Zawawi
Hasni Amiruddin
Farizan Othman
Dept. of Industrial Computing
Faculty of Technology and Information Science
Universiti Kebangsaan Malaysia
43600 UKM Bangi
Selangor Darul Ehsan
Malaysia
yy@ftsm.ukm.my
This project presents Interactive Learning-Mathematica Enhance Calculus (ILMEC), a multimedia courseware that was developed using Mathematica as the computational kernel. ILMEC serves three roles in the teaching and learning of high school calculus in Malaysia, i.e., it helps in enhancing the understanding of mathematical concept and skill, mathematical problem solving and mathematical reasoning.
In the realm of mathematical problem solving, ILMEC provides (i) the ability to focus on the process of problem solving and not just on the computational aspect [1] and (ii) aid on solving realistic problems that involve large numbers and not restricted to contrived problems having nice solutions [2]. The method of problem solving used in ILMEC is called a step-by-step solution method. In this paper, the method is applied on the pre calculus topics such as geometry coordinate, quadratic equation, quadratic function and simultaneous equation. Multimedia technology is also used to provide a stimulating environment during the process of problem solving. It encourages exploration, self-expression and a feeling of ownership by allowing students to manipulate its components and makes learning stimulating, engaging and fun [3]. The exercises are developed by applying multimedia technology and linked to Mathematica in order to make them interactive and enjoyable. Students are also able to experiment by putting different values in the input box during a problem solving process. Apart from that, graphic facilities that are generated by Mathematica are used extensively to enrich the student s understanding of a problem.
Deborah Hughes Hallett,
Professor of Mathematics,
University of Arizona;
Adjunct Professor,
Kennedy School of Government, Harvard
Computers are changing the teaching and practice of mathematics. Numerical methods are now as easy to perform as symbolic calculations; visualization and graphical reasoning have become much more common. In addition, simulations allow us to bring experimental evidence into the mathematics classroom. This can shift the way in which mathematics is perceived from a subject in which the rules are given to a subject in which the rules are obtained by observing mathematical phenomena. Making use of simulations in teaching requires both new materials and different pedagogy.
This talk will describe two examples of simulation for elementary courses, one on medical testing and one on bidding strategy in an auction. One simulation models a process that the students can do analytically, though not with ease. The other simulation models a process that students at the introductory level cannot do analytically. We will contrast the roles of the two simulations in the learning process, and consider under what circumstances it is appropriate to use a simulation as an introduction to an analytical solution and in what circumstances it is appropriate to use a simulation without an analytical solution.
The examples come from a freshman level mathematics for business students at the University of Arizona, and a mathematics review course for midcareer public servants at the Kennedy School of Government, Harvard.